Castedo Ellerman

Entropy Scope of Relevance

Proposed answer to the following questions:

Related questions:

Let $$\operatorname{dom}{ f}$$ denote the domain of a function $$f$$. Given $$\pi$$ a set of sets, let $${\cup{ \pi}}$$ denote the union of all member sets in $$\pi$$. When $$\pi$$ is a partition, $$\pi$$ covers $${\cup{ \pi}}$$.

In a scope of relevance, partitions represent questions with each member set (part) an answer. Each answer represents an event of a probability space.

Given a function over partitions of events of probability space $$\Omega$$, define levels of the domain:

$\begin{eqnarray*} \operatorname{dom}_{ 0}{ \theta} & := & \{ \pi \in \operatorname{dom}{ \theta} : {\cup{ \pi}} = \Omega \} \\ \operatorname{dom}_{ i+1}{ \theta} & := & \left\{ \pi \in \operatorname{dom}{ \theta} : {\cup{ \pi}} \in \rho, \rho \in \operatorname{dom}_{ i}{ \theta} , \pi \not\in \operatorname{dom}_{ i}{ \theta} \right\} \\ \end{eqnarray*}$

A scope of relevance $$\theta$$ is a non-negative real-valued function over partitions of events (subsets) of a probability space $$\Omega$$ with the following conditions:

1. At most one partition can cover any event. Formally, $${\cup{ \pi}} = {\cup{ \rho}}$$ implies $$\pi = \rho$$ for any partitions $$\pi$$ and $$\rho$$ in $$\operatorname{dom}{ \theta}$$.
2. $$\operatorname{dom}{ \theta} = \bigcup_{i=0}^\infty \operatorname{dom}_{ i}{ \theta}$$

The domain of a scope of relevance is all relevant questions. Each real value assigned to a question is a degree of relevance. The second condition in the definition means the partitions (questions) are dividing the probability space $$\Omega$$ in a nested hierarchical manner. Or in other words, either the event of all outcomes ($$\Omega$$) or the event of an answer to a question is the event under which another question can be asked.